Hostname: page-component-cb9f654ff-plnhv Total loading time: 0.001 Render date: 2025-08-30T21:23:51.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Zero-one dual characters of flagged Weyl modules

Part of: Algebraic combinatorics Projective and enumerative geometry Combinatorics

Published online by Cambridge University Press:  08 July 2025

Peter L. Guo ,
Zhuowei Lin  and
Simon C. Y. Peng
Peter L. Guo*
Affiliation:
Center for Combinatorics, https://ror.org/01y1kjr75 Nankai University , LPMC, Tianjin 300071, P.R. China
Zhuowei Lin
Affiliation:
Center for Combinatorics, https://ror.org/01y1kjr75 Nankai University , LPMC, Tianjin 300071, P.R. China e-mail: zwlin0825@163.com
Simon C. Y. Peng
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, P.R. China e-mail: pcy@tju.edu.cn
*
e-mail: lguo@nankai.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

Keywords

Flagged Weyl moduledual characterSchubert polynomialkey polynomialzero-one polynomial

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory 05A19: Combinatorial identities, bijective combinatorics 14N15: Classical problems, Schubert calculus

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 23
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Armon, S., Assaf, S., Bowling, G., and Ehrhard, H., Kohnert’s rule for flagged Schur modules . J. Algebra 617(2023), 352–381.Google Scholar
Assaf, S. and Searles, D., Kohnert polynomials . Exp. Math. 31(2022), no. 1, 93–119.Google Scholar
Assaf, S. and Searles, D., Kohnert tableaux and a lifting of quasi-Schur functions . J. Combin. Theory Ser. A 156(2018), 85–118.Google Scholar
Brion, M., Multiplicity-free subvarieties of flag varieties . In: Commutative algebra (Grenoble/Lyon, 2001), Contemporary Mathematics, 331, American Mathematical Society, Providence, RI, 2003, pp. 13–23.Google Scholar
Castillo, F., Cid-Ruiz, Y., Mohammadi, F., and Montaño, J., K-polynomials of multiplicity-free varieties. Preprint, 2022. arXiv:2212.13091.Google Scholar
Eur, C. and Larson, M., K-theoretic positivity for matroids. Preprint, 2023. arXiv:2311.11996.Google Scholar
Fan, N. J. Y. and Guo, P. L., Vertices of schubitopes . J. Combin. Theory Ser. A 177(2021), 105311, 20pp.Google Scholar
Fan, N. J. Y. and Guo, P. L., Upper bounds of Schubert polynomials . Sci. China Math. 65(2022), 1319–1330.Google Scholar
Fink, A., Mészáros, K., and St, A., Dizier, Schubert polynomials as integer point transforms of generalized permutahedra . Adv. Math. 332(2018), 465–475.Google Scholar
Fink, A., Mészáros, K., and St Dizier, A., Zero-one Schubert polynomials . Math. Z. 297(2021), 1023–1042.Google Scholar
Gao, Y., Hodges, R., and Yong, A., Classification of Levi-spherical Schubert varieties . Selecta Math. (N.S.) 29(2023), Paper No. 55. 40 pp.Google Scholar
Hamaker, Z., Pechenik, O., and Weigandt, A., Gröbner geometry of Schubert polynomials through ice . Adv. Math. 398(2022), Paper No. 108228. 29 pp.Google Scholar
Hodges, R. and Yong, A., Coxeter combinatorics and spherical Schubert geometry . J. Lie Theory 32(2022), 447–474.Google Scholar
Hodges, R. and Yong, A., Multiplicity-free key polynomials . Ann. Comb. 27(2023), 387–411.Google Scholar
Klein, P., Diagonal degenerations of matrix Schubert varieties . Algebraic Combin. 6(2023), 1073–1094.Google Scholar
Knutson, A., Lam, T., and Speyer, D. E., Positroid varieties: Juggling and geometry . Compositio Math. 149(2013), 1710–1752.Google Scholar
Kohnert, A., Weintrauben, Polynome, Tableaux . Bayreuth Math. Schrift. 38(1990), 1–97.Google Scholar
Kraskiewicz, W. and Pragacz, P., Foncteurs de Schubert . C. R. Acad. Sci. Paris Sér. I Math. 304(1987), 209–211.Google Scholar
Kraśkiewicz, W. and Pragacz, P., Schubert functors and Schubert polynomials . Eur. J. Combin. 25(2004), 1327–1344.Google Scholar
Knutson, A., Frobenius splitting and Möbius inversion. Preprint, 2009. arXiv:0902.1930v1.Google Scholar
Knutson, A. and Miller, E., Gröbner geometry of Schubert polynomials . Ann. Math. 161(2005), 1245–1318.Google Scholar
Magyar, P., Schubert polynomials and Bott–Samelson varieties . Comment. Math. Helv. 73(1998), 603–636.Google Scholar
Mészáros, K., Setiabrata, L., and St Dizier, A., An orthodontia formula for Grothendieck polynomials . Trans. Amer. Math. Soc. 375(2022), 1281–1303.Google Scholar
Mészáros, K., Setiabrata, L., and St Dizier, A., On the support of Grothendieck polynomials . Ann. Comb. 29(2025), no. 2, 541–562.Google Scholar
Mészáros, K., Dizier, A. St., and Tanjaya, A., Principal specialization of dual characters of flagged Weyl modules . Electron. J. Combin. 28(2021), Paper No. 4.17. 12 pp.Google Scholar
Mészáros, K. and Tanjaya, A., Inclusion-exclusion on Schubert polynomials . Algebr. Comb. 5(2022), 209–226.Google Scholar
Pechenik, O. and Satriano, M., Proof of a conjectured Möbius inversion formula for Grothendieck polynomials . Selecta Math. (N.S.) 30(2024), Paper No. 83. 8 pp.Google Scholar
Peng, S. C. Y., Lin, Z., and Sun, S. C. C., Upper bounds of dual flagged Weyl characters . Adv. Appl. Math. 160(2024), 102752, 12pp.Google Scholar
Postnikov, A., Total positivity, Grassmannians, and networks. Preprint, 2006. http://arxiv.org/abs/math/0609764.Google Scholar